A radioactive substance decays exponentially. A scientist begins with 130 milligrams of a radioactive substance. After 20 hours, 65 mg of the substance remains. How many milligrams will remain after 24 hours?
half of the substance remains after 20 hours ... so that is the half-life
after 24 hr ... amount = 130 * (1/2)^(24 / 20)
Well, if the substance is decaying exponentially, it's certainly feeling a bit "radioactive" about it! Now, let's address the question at hand.
We know that after 20 hours, 65 mg of the substance remains, indicating that half of the substance has decayed. So, we can say that after 20 hours, we have 65 mg, and after 10 hours, we would have half of that — 32.5 mg.
Now, let's see what happens after 24 hours. Since the substance decays exponentially, we can assume that it would be half of 32.5 mg after another 10 hours.
However, since 20 hours have already passed, we only need to consider the remaining 4 hours. So taking half of 32.5 mg, we would have 16.25 mg after 24 hours.
So, after 24 hours, approximately 16.25 mg of the radioactive substance will remain. Remember, this is just an estimate, so don't let those radioactive elements play any practical jokes on you!
To solve this problem, we can use the exponential decay formula:
A = A0 * e^(-kt)
Where:
A = Final amount of the radioactive substance
A0 = Initial amount of the radioactive substance
k = Decay constant
t = Time in hours
We can calculate the decay constant (k) using the information provided:
65 = 130 * e^(-20k)
Dividing both sides of the equation by 130, we get:
0.5 = e^(-20k)
Taking the natural logarithm of both sides, we have:
ln(0.5) = -20k
Now, we can solve for k:
k = ln(0.5) / -20
Calculating the value, we find:
k ≈ 0.035
Now, we can use the calculated value of k to determine the amount of the radioactive substance after 24 hours:
A = 130 * e^(-0.035 * 24)
Calculating the value, we find:
A ≈ 55.17
Therefore, approximately 55.17 milligrams will remain after 24 hours.
To solve this problem, we need to use the formula for exponential decay:
N(t) = N₀ * e^(-kt),
where:
N(t) is the quantity of the substance remaining at time t,
N₀ is the initial quantity of the substance,
e is the base of the natural logarithm (approximately 2.71828),
k is the decay constant, and
t is the time elapsed.
We are given that the initial quantity is 130 mg (N₀ = 130) and the quantity remaining after 20 hours is 65 mg (N(20) = 65). We need to find N(24).
Let's plug in the given values into the formula to find the decay constant (k):
65 = 130 * e^(-20k).
Divide both sides by 130:
0.5 = e^(-20k).
To find k, we can take the natural logarithm (ln) of both sides:
ln(0.5) = -20k.
Divide both sides by -20:
k = ln(0.5) / -20.
Now that we have the decay constant (k), we can calculate N(24):
N(24) = 130 * e^(-24k).
Plug in the value of k we found earlier:
k = ln(0.5) / -20,
N(24) = 130 * e^(-24 * (ln(0.5) / -20)).
Using a calculator or software, we can evaluate this expression to find the approximate value of N(24). After calculating, we find that approximately 59.865 mg of the substance will remain after 24 hours.